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So In my final year of my Undergraduate Degree (Studying Mathematics and Economics) I have decided to focus my Undergraduate project on Game Theory.

I have done quite a bit of research, and I get the basics of different strategy types, 2xn games, and certain solutions, some key results and proofs by John Nash.

However the project itself that my University requires, does not ask to reply to a question, it's main goal is to present research on the topic. However, what is important it is still a Mathematical paper, rather than an Economic one, therefore it is to be more "Mathematically advanced" than if writing and Economics paper.

My question is:

Would you have any suggestions for some extensive fields,interest within Mathematical Game Theory?

Do you know of any examples of applications of other fields of mathematics (e.g.g algebraic geometry) to game theory, or the other way around? (e.g. the Nash's proof for equilibrium in n-played non-cooperative non-zero sum games using the fixed point theorem)?

Perhaps any sources for application of game theory to AI? (Of which I have heard a lot of, but I can't find any good sources)

It's a huge field, and I'm looking for places where i can narrow down my project.

Regards Mat

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A couple ideas. First, Knuth's Surreal Numbers can be interpreted as games, so that is an interesting math/game match.
Some more advanced math arises in questions of repeated games. For example, how to value a stream of payoffs into the infinite future? A natural way for economists to do it is by discounting the future, but other people have used the overtaking criterion and even something called Banach Limits (Mailath and Samuelson, repeated games and reputations chapter 3). Also in repeated games, finding the right fixed point theorem to use is non-trivial. Ky Fan's is often used. From the economics side, there is interest in finite autonoma playing games, but whether the computer science people have picked up on that I don't know. Try chapter 9 of Osborne and Rubinstein. Maybe this will inspire others who know more to join in.

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